WASHINGTON—President Trump threatened again Friday to shut down all or a portion of the U.S. southern border as administration officials said that the influx of illegal immigrants there was straining resources beyond the breaking point.

While touring Florida’s Lake Okeechobee Friday, Mr. Trump called on Mexico to stop asylum-seeking families from reaching the U.S. border. “We will keep the border closed,” he said. “And we will keep it stopped for a long time. I’m not playing games.”

Homeland Secretary
Kirstjen Nielsen
Friday said her agency expected that as many as 100,000 families, unaccompanied children and other migrants would cross the border illegally or ask for refuge at legal border crossings this month. Last month, about 66,000 people were arrested as they crossed the border illegally, more than half of whom were families or unaccompanied children. There is a “cascading crisis at our southern border. The system is in free fall,” said Ms. Nielsen.

Ms. Nielsen sent a letter to Congress on Thursday repeating requests that Congress overhaul immigration laws. Trump administration officials have described the law as full of legal loopholes that both attract migrants to the U.S. and make it difficult to jail families or swiftly deport them if they aren’t eligible for asylum.

Mr. Trump’s comments Friday come as part of his broader push against Democrats seeking to build momentum for the 2020 election following the conclusion of special counsel Robert Mueller’s investigation. Attorney General
William Barr
has said Mr. Mueller’s report determined Mr. Trump and his campaign didn’t collude with the Russian government in the 2016 election.

“Democrats want to pretend there is no border crisis for one simple reason, because they have caused the border crisis, and you know why else honestly, because they want votes,” he said at a rally in Michigan Thursday.

Last year, Mr. Trump threatened to close the border when migrant caravans made their way north from Central America, but it didn’t happen. Since then, other caravans have passed through Mexico, and many of the migrants have joined lengthy lines of people waiting to ask for asylum at legal border crossings.

Tens of thousands of migrants are crossing the border illegally every month to ask for asylum. From the start of the federal budget year in October through February, more than 136,000 people traveling in families have been arrested crossing the border illegally, a tally higher than the previous 12-month record set during fiscal 2018.

A senior Department of Homeland Security official said Friday that Mr. Trump was stating his intention if necessary to close some ports of entry—places like El Paso, Texas, Yuma, Ariz., and San Diego.

“It’s on the table,” the official said. “I will make a recommendation accordingly to the president.” The official said the U.S. had made additional proposals to Mexico. “We have walked through a whole variety of things they can do for enforcement.”

Victor Manjarrez, a former Border Patrol sector chief who spent part of his career in El Paso, said closing the border would almost certainly mean sealing legal border crossings. Doing that, he said, would have little immediate impact on illegal border crossers, but it might force the Mexican government to more forcefully block people from crossing its southern border to alleviate potential crowding in the country’s north.

Such a closure could also strand U.S. citizens who got stuck on the wrong side of a crossed border.

Border Patrol agents have been temporarily redeployed from the Canadian border and inland highway checkpoints, along with officers assigned to legal border crossings, to help manage the detention and processing of migrants in places such as El Paso. The numbers there were so overwhelming this week that some migrants were temporarily held under a bridge outside a Border Patrol station.

Immigration officials only have space to house about 3,000 migrant family members for up to 20 days each, which isn’t long enough for their immigration cases to be decided. With a backlog of more than 855,000 cases in federal immigration court, it can take years for a case to be decided. As of Friday, just more than 1,000 immigrant family members were being held.

Immigration and Customs Enforcement said Friday that transportation and other processing costs associated with the families’ arrival has left them unable to keep those jails filled with people who can only be held for about three weeks.

DHS officials said that agency was also overwhelmed by the arrival of more than 26,000 unaccompanied children in the last five months. Children are spending days in Border Patrol custody in temporary holding facilities never intended to hold anyone for more than a few hours at a time.

After meeting this week with Ms. Nielsen, Mexican Interior Minister Olga Sánchez said Mexico would no longer give out humanitarian visas en masse, but will grant Central Americans temporary visitor and work visas for the southeast of the country.

She said the aim was to contain the northbound flow of migrants with the installation of migration, federal police and civil-protection stations at the Isthmus of Tehuantepec, a 125-mile wide stretch of the country between the Gulf and Pacific coasts.

“We’re not going to militarize our southern border, let that be clear, and I expressed that to [Secretary] Nielsen,” she told reporters. She added that the governments of Central America needed to help. “It’s not only the U.S., it’s not only Mexico,” she told reporters.

Mexico has been deporting tens of thousands of Central Americans in the last year. The International Organization for Migration, a United Nations agency, said that between January and August 2018, Mexico deported 65,210 Central Americans, while U.S. authorities returned 64,239 people to Central America during the same time.

Mexico increased such deportations by about 40% compared with the same period in 2017, the agency’s data shows.

Most of the families are from Guatemala, Honduras and El Salvador and say they fleeing poverty, violence and political instability.